FAD1015: Distribution Reading Note — Poisson, Normal, Uniform, Exponential

Purpose: Build intuition for what each distribution IS, when it shows up, and how to tell them apart at a glance. Zero formulas — just cues and concepts.

Poisson Distribution (Discrete)

What it is

Poisson counts how many times something happens in a fixed window — a stretch of time, an area of land, a volume of liquid. It is the only distribution where the mean equals the variance — if those two numbers are the same, your data is very likely Poisson.

Shape

Skewed right when lambda is small (most of the mass near zero, long tail to the right). As lambda grows, it becomes more symmetric and starts looking like a bell curve — but it still lives on whole numbers only.

Support

Non-negative integers: 0, 1, 2, 3, ... (no upper bound — unlike Binomial which is capped at n).

Signature trait

Mean and variance are identical. If mean does not equal variance, it is not Poisson.

Cue phrases — when to reach for Poisson

  • "Number of [events] per [time unit / area / volume]"
  • "Occurs at an average rate of ... per hour / day / week"
  • "Random and independent occurrences"
  • "Rare events" — calls to a help centre, accidents at an intersection, defects on a production line
  • The problem gives a rate per time period but asks about a different time period (you adjust the rate — this is a common exam move)

Common exam traps

  • Forgetting to scale the rate — if lambda = 3 per week and the problem asks about 2 weeks, the new lambda is 6, not 3
  • Using Poisson when the events are not independent — e.g. one accident causing another nearby
  • Assuming Poisson when variance clearly differs from the mean — if mean is 5 but variance is 12, this is not Poisson

Connections

  • Poisson to Binomial: When n is large and p is small (or large), Binomial can be approximated as Poisson
  • Poisson to Exponential: They are two sides of the same coin. Poisson counts how many events in a window; Exponential measures how long between consecutive events. Same underlying process, different question.

Normal Distribution (Continuous)

What it is

The bell curve. Symmetric, unimodal, and everywhere in statistics. Most real-world measurements that are the sum of many small independent factors end up normally distributed. It is the workhorse of statistical inference.

Shape

Perfectly symmetric bell. Single peak in the centre. Tails extend infinitely in both directions but get arbitrarily close to zero — never touch the axis. The centre is simultaneously the mean, median, and mode (all three are the same point).

Support

All real numbers: negative infinity to positive infinity. Every value on the number line is possible, but values far from the centre are extremely unlikely.

Signature trait

The 68-95-99.7 rule: about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. If a problem describes a "bell-shaped" or "symmetric" distribution, think Normal.

Cue phrases — when to reach for Normal

  • "Bell-shaped", "symmetric", "approximately normal"
  • "Mean = median = mode"
  • "Standard deviation" or "variance" given alongside a mean for a natural measurement
  • Test scores, heights, weights, blood pressure, measurement errors
  • The problem says "assume normality" — this is the biggest freebie
  • A sample size of 30 or more — the Central Limit Theorem kicks in and the sampling distribution of the mean is approximately Normal regardless of the original population shape
  • "Standardize" or "z-score" — these are Normal-specific operations
  • Using the empirical rule to estimate proportions without complex calculations

Common exam traps

  • Forgetting to square root the variance when standardizing — the Normal's second parameter can be given as variance (sigma squared) or standard deviation (sigma), and mixing them up changes all answers
  • Reading the z-table backwards — most tables give the right-tail area P(Z > z), not P(Z < z)
  • Applying the empirical rule to non-Normal data — it only works for true bell curves
  • Not doing a continuity correction when using Normal to approximate Binomial — the 0.5 adjustment is easy to forget and costs marks

Connections

  • Normal to Binomial: When np and nq are both large (typically > 5), Binomial can be approximated as Normal (with continuity correction)
  • Normal to Sampling distributions: The Normal distribution is why confidence intervals and hypothesis tests work — thanks to the Central Limit Theorem

Uniform Distribution (Continuous)

What it is

Every value in the interval is equally likely. The "flat" distribution. If you pick a random number between 0 and 1 on a calculator, that is Uniform. There is no peak, no skew — everything is the same height.

Shape

Perfectly flat rectangle. Constant height from a to b, zero everywhere else.

Support

Any real number between a and b inclusive. Nothing outside the interval is possible.

Signature trait

The PDF is a constant — no shape, no curve. If the probability is the same for all outcomes in the range, it is Uniform.

The CDF is a straight diagonal line from (a,0) to (b,1). To find the probability that X is less than any value k, you compute:

$$\text{Fraction of the interval from a to k} = \frac{k-a}{b-a}$$

This is the core calculation for every Uniform probability problem — you are just measuring how far k is from a as a proportion of the total interval length. For example:

  • P(X < 15) on U(10, 30) = (15-10)/(30-10) = 5/20 = 0.25
  • P(X > 25) on U(10, 30) = (30-25)/(30-10) = 5/20 = 0.25 (flip to right tail)
  • P(12 < X < 18) on U(10, 30) = (18-12)/(30-10) = 6/20 = 0.30 (subtract the two CDF values)

No integration, no tables — just (target interval length) / (total interval length).

Cue phrases — when to reach for Uniform

  • "Randomly selected between ... and ..."
  • "Equally likely outcomes over an interval"
  • "Completely random time" — e.g. a bus arrives at any time within a 10-minute window with equal probability
  • "Uniform distribution" explicitly stated
  • The probability is simply length of the desired interval divided by total length — no curves, no integration tricks
  • Rounding errors, random number generation, waiting times where no time is preferred over any other

Common exam traps

  • Confusing Uniform with Normal — both are continuous, but Normal has a peak and tapers off; Uniform is flat. If the problem says "bell-shaped", it is Normal. If it says "any time equally likely", it is Uniform.
  • Forgetting the mean formula — since the distribution is symmetric, the mean is always the midpoint.
  • Mixing up the standard deviation formula — the denominator is sqrt(12), not sqrt(2) or anything else

Connections

  • Uniform is the simplest continuous distribution — it serves as a building block for generating random numbers in simulations
  • It does NOT approximate or get approximated by any other distribution in FAD1015 — it stands alone as the "flat" baseline

Exponential Distribution (Continuous)

What it is

The waiting time distribution. It models how long you wait until the next event happens, assuming events occur at a constant average rate and independently of each other. It is the continuous counterpart to the Poisson.

Shape

Starts high at x=0 and decays exponentially downwards — a steep drop that gradually flattens out. Most of the probability mass is near zero (short waiting times are most likely). Long waiting times are possible but increasingly unlikely.

Support

Non-negative real numbers: 0 to positive infinity. Cannot be negative — you cannot wait a negative amount of time.

Signature trait

Memoryless. The time already spent waiting does not matter. If you have waited 10 minutes for a bus and the average wait is 5 minutes, the expected remaining wait is still 5 minutes — not 0, not 10. This is unique to Exponential. No other distribution has this property.

Also unique: the mean equals the standard deviation (the reciprocal of the rate).

Cue phrases — when to reach for Exponential

  • "Time until / time between / waiting time for the next [event]"
  • "Lifespan" or "lifetime" of a component that does not age (like a light bulb or a computer chip)
  • "Service time" — how long a customer takes at a checkout counter
  • The problem says "exponential" explicitly, or mentions a Poisson process and asks about waiting time
  • "Memoryless" — if this word appears, it can only be Exponential
  • "Average time between ... is ... minutes / hours"

Common exam traps

  • Confusing Exponential with Poisson: Poisson = number of events; Exponential = time between events. A Poisson process generates both — you just flip which side of the coin you look at.
  • Forgetting the memoryless property — you cannot say "the battery has already lasted 80 hours so it is more likely to die soon." Exponential says it is not. The probability of lasting another 50 hours is the same regardless of whether it is brand new or 80 hours old.
  • Mixing up the rate and the mean — lambda is the rate (events per unit time), while 1/lambda is the mean waiting time. If lambda = 5 per hour, the average wait is 1/5 hour = 12 minutes.
  • Not adjusting the rate when changing time units — lambda must be in the same units as the question

Connections

  • Exponential to Poisson: Inseparable. If events follow a Poisson process, the counts per interval are Poisson and the waiting times are Exponential. Same lambda, just different questions: "how many?" vs "how long?"
  • Exponential does not approximate or get approximated by any other distribution in FAD1015

At a Glance: Cue-Phrase Cheat Sheet

Scenario in problem Distribution Why
"Counts the number of events per hour/day/week" Poisson Counting in a fixed window
"Mean equals variance" Poisson Only distribution where this is true
"Random and independent occurrences over time" Poisson Standard Poisson conditions
"Rate is given, adjust for different time period" Poisson Rate scaling is a classic exam step
"Bell-shaped, symmetric" Normal Classic Normal description
"Assume normality" Normal Direct instruction
"Mean = median = mode" Normal Only symmetric unimodal continuous
"68%, 95%, 99.7%" Normal Empirical rule
"Standardize" or "z-score" Normal Only Normal gets standardized
"Sample size 30 or more" Normal CLT makes the sampling dist Normal
"Every value equally likely" Uniform Flat = Uniform
"Randomly selected between a and b" Uniform Equal probability across range
"Any time equally likely" Uniform Flat waiting time
"No preference for any value" Uniform Equal likelihood = Uniform
"Time until the next event" Exponential Waiting time for next occurrence
"Time between events" Exponential Continuous gap between Poisson events
"Memoryless" or "does not age" Exponential Unique property
"Lifespan / lifetime of a component" Exponential Memoryless failure rate
"Mean = standard deviation" Exponential Also unique to Exponential
"Service time at a counter" Exponential Waiting time in a queue

How They Connect

Poisson process (events at constant average rate, independent):

  • Poisson: "How many events in the next hour?" (discrete count)

  • Exponential: "How long until the next event?" (continuous time)

  • Poisson to Normal: For large lambda, Poisson looks almost bell-shaped and can be approximated by a Normal distribution

  • Poisson to Binomial: When n is large and p is small, Binomial to Poisson. When both np and nq are large, Binomial to Normal.

  • Normal and Uniform are not directly connected — they are the two extremes of continuous distributions

  • Exponential and Uniform share the waiting-time context but are fundamentally different

How to Choose: A Decision Tree

Start at the top and follow the yes/no answers.

Is the variable discrete (counts) or continuous (measurements)?

Discrete:

  • Does it count events in a fixed window of time/space/volume?
    • Poisson
  • Does it count successes in a fixed number of trials?
    • Binomial (not covered here)

Continuous:

  • Is the probability the same for every value in the range?
    • Uniform
  • Is it bell-shaped and symmetric?
    • Normal
  • Is it measuring the time until the next event?
    • Exponential

Decision by Exam Question Pattern

The problem gives you... And asks you to... Use
A rate (e.g. 4 per hour) and a time period Find probability of exactly x events Poisson
A rate and asks "what is the chance we see 0 events?" Plug into Poisson with x=0 Poisson
Mean and standard deviation of a measurement Find probability above/below a value Normal
Sample mean, sample size, and population parameters Find probability that the sample mean exceeds a value Normal (CLT)
Two numbers a and b Find probability of a value between them Uniform
An interval and a statement "equally likely" Any probability calculation Uniform
A rate and asks "probability the next event occurs within t minutes" Exponential CDF Exponential
A lifetime with constant failure rate Probability of surviving past time t Exponential
A component that has already survived some time Conditional probability using memoryless property Exponential

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